Reentrant Fulde-Ferrell-Larkin-Ovchinnikov superfluidity in the honeycomb lattice

We study superconducting properties of population-imbalanced ultracold Fermi mixtures in the honeycomb lattice that can be effectively described by the spin-imbalanced attractive Hubbard model in the presence of a Zeeman magnetic field. We use the mean-field theory approach to obtain ground-state phase diagrams including the unconventional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, which is characterized by atypical behavior of the Cooper-pair total momentum. We show that the momentum changes its value as well as direction with a change of the system parameters. We discuss the influence of Van Hove singularities on the possibility of the reentrant FFLO phase occurrence, without a BCS precursor.

Figure 1

(a) Honeycomb lattice discussed in this paper. The unit cell defined by vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$ containing two atoms (blue and green points) belonging to sublattices $A$ and $B$. The three nearest-neighbor vectors are given by ${\mathbf{\delta }}_1=(\sqrt{3}/2,-1/2), {\mathbf{\delta }}_2=(0,1)$, and ${\mathbf{\delta }}_3=(-\sqrt{3}/2,-1/2)$. The elementary primitive unit cell (yellow rhombus) is given by lattice vectors ${\mathit{a}}_1={\mathbf{\delta }}_2-{\mathbf{\delta }}_3=a(1/2,\sqrt{3}/2)$ and ${\mathit{a}}_2={\mathbf{\delta }}_2-{\mathbf{\delta }}_1=a(-1/2,\sqrt{3}/2)$, where $a=\sqrt{3}$. (b) Dispersion relation for $\mu /t=0$ and $h/t=0$. Reciprocal lattice vectors are given by ${\mathit{b}}_1=2\pi /a(1,1/\sqrt{3})$ and ${\mathit{b}}_2=2\pi /a(-1,1/\sqrt{3})$, while the first Brillouin zone is shown by a black dotted hexagon. High symmetry points are given by $M={\mathit{b}}_1/2=2\pi /a(1/2,1/2\sqrt{3})$ and $K=-K^{\prime }=({\mathit{b}}_1-{\mathit{b}}_2)/3=2\pi /a(2/3,0)$.

Figure 2

Energy band structure of the honeycomb lattice. The FBZ is shown by the black hexagon. Simultaneously, the hexagon shows the Fermi level in the case of half filling ($\mu /t=0$) and absence of the magnetic field ($h/t=0$). The Dirac cones are located in the corner $K$ and $K^{\prime }$ points of the FBZ.

Figure 3

Amplitude of the superconducting order parameter in the absence of the Zeeman magnetic field as a function of the chemical potential $\mu$ and pairing interaction $U$.

Figure 4

Amplitude of the superconducting order parameter ${\mathrm{\Delta }}_0$ as a function of the chemical potential $\mu$ and attractive interaction $U$. The results (a) in the absence ($h=0.0t$) and (b) in the presence ($h=1.0t$) of the magnetic field $h$. Stars in (b) show the phase transition from FFLO to BCS with increasing $\left|U\right|$.

Figure 5

Ground-state phase diagram; the magnetic field $h$ vs the attractive interaction $U$ at (a) $\mu /t=0$ and (b) $\mu /t=\pm 1$. The colormap shows the SOP amplitude ${\mathrm{\Delta }}_0$ (blue and red for BCS and FFLO phases, respectively), whereas white indicates the normal (NO) phase. At $\mu /t=0$ ($n=1$), the reentrant FFLO superconductivity is stable around $h/t=1$, below $U_c$.

Figure 6

Ground-state phase diagram: magnetic field $h$ vs chemical potential $\mu$ for different values of the interaction $U$: (a) $U=-2.0t$, (b) $U=-2.5t$, (c) $U=-3.0t$, and (d) $U=-3.5t$. The colormap shows the SOP amplitude ${\mathrm{\Delta }}_0$ (blue and red for BCS and FFLO phases, respectively, and white for the NO phase). Green dashed lines indicate parameters for which the VHSs are located at the Fermi level.

Figure 7

Ground-state phase diagram: magnetic field $h$ vs average number of particles $n$ for (a) $U=-2.0t$, (b) $U=-2.5t$, (c) $U=-3.0t$, and (d) $U=-3.5t$. The colormap shows the SOP amplitude ${\mathrm{\Delta }}_0$ (blue and red for BCS and FFLO phases, respectively, white for the NO phase). The yellow area denotes the phase separation region. Green dashed lines indicate parameters for which the VHSs are located at the Fermi level.

Figure 8

The DOS of the honeycomb lattice ($U=0$) for different values of the chemical potential $\mu$ and external magnetic field $h$. Red and blue lines denote the DOS for particles with spin $\uparrow$ and $\downarrow$, respectively, whereas the Fermi level is shown as gray line. The scheme for the different parameters is as follows: (a) $\mu /t=0$ and $h/t=0$, (b) $\mu /t=0$ and $h/t=1$, (c) $\mu /t=1$ and $h/t=0$, and (d) $\mu /t=1$ and $h/t=2$.

Figure 9

Fermi surface of the honeycomb lattice for different values of the Fermi level shown by isoenergetic red lines. The first Brillouin zone is shown by a blue hexagon. Vectors show an example of the Cooper-pair formation, while solid red line shows the Fermi level for fillings equal to 3/8 and 5/8.

Figure 10

Spatial decomposition of the SOP in real space for different total momenta of Cooper pairs $\mathit{Q}$ [marked by points $b$–$g$ in (a)]. The color (red and blue) and the size of the circles correspond to the sign ($+$ and $-$) and the value of the order parameter. White lines are nodal lines in real space.

Figure 11

(a) Mean-field ground-state phase diagram. The magnetic field $h$ vs pairing interaction $U$ at half filling ($\mu /t=0$). In the case when the FFLO phase is neglected in calculations, the dashed red line indicates the critical magnetic field above which the BCS state is unstable. Above this line, the normal state (NO) exists. The occurrence of the FFLO state in the phase diagram slightly shifts the boundary of the BCS phase, which is indicated by the solid blue line. The labels $A, B$, and $C$ show three different directions of the total momentum $\mathit{Q}$ for which the FFLO phase is realized. (b) Schematic picture of vectors $\mathit{Q}$ in the FBZ, for three different variants of the FFLO phase: $A, B$, and $C$.

Figure 12

Components of the Cooper-pair momentum vector $\mathit{Q}=(Q_x,Q_y)$ as a function of the interaction strength $U$ and the external magnetic field $h$.